These twelve questions are original and cover the whole module: negative and fractional indices, standard form, comparing sizes, simplifying roots and estimating. They run from easy to harder.
Work each one on paper first, without a calculator unless the question says otherwise. Then open the answer and compare your working, not just your result. Note any question you miss in the mistake log, so you can retest it in a few days.
Indices
Question 1. Find the value of 3−2.
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A negative power means a reciprocal. 3−2 = 1/32 = 1/9.
Question 2. Simplify (25 × 2−2) ÷ 2−1, giving your answer as a number.
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Multiply: 25 × 2−2 = 25+(−2) = 23. Divide: 23 ÷ 2−1 = 23−(−1) = 24 = 16.
Check: 8 ÷ (1/2) = 16.
Question 3. Find the value of 163/4.
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The denominator 4 means the fourth root: the fourth root of 16 is 2, since 24 = 16. The numerator 3 means cube it: 23 = 8.
Question 4. Find n if 3n = 1/81.
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81 = 34, so 1/81 = 3−4. Therefore n = −4.
Standard form
Question 5. Write 0.000036 in standard form.
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The decimal point moves 5 places to sit after the 3, giving 3.6. The number is small, so the power is negative: 3.6 × 10−5.
Check: 3.6 × 10−5 is 3.6 hundred-thousandths, which is 0.000036.
Question 6. Work out (3 × 10−4) × (5 × 107), giving your answer in standard form.
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Multiply the numbers: 3 × 5 = 15. Add the powers: −4 + 7 = 3. So the product is 15 × 103. This is not yet standard form, because 15 is bigger than 10. Write 15 = 1.5 × 101, so the answer is 1.5 × 104.
Check: 0.0003 × 50 000 000 = 15 000.
Question 7. Write in ascending order: 4.5 × 10−2, 0.0062, 3.9 × 10−3.
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Write each as an ordinary number: 0.045, 0.0062 and 0.0039. The smallest is 0.0039, then 0.0062, then 0.045. In the original forms: 3.9 × 10−3, 0.0062, 4.5 × 10−2.
Roots
Question 8. Simplify √108.
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The largest square factor of 108 is 36, because 108 = 36 × 3. So √108 = √36 × √3 = 6√3.
Check: 6√3 squared is 36 × 3 = 108.
Question 9. Simplify √48 − √12.
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√48 = √(16 × 3) = 4√3. √12 = √(4 × 3) = 2√3. Then 4√3 − 2√3 = 2√3.
Check: 2√3 ≈ 3.46, and √48 − √12 ≈ 6.93 − 3.46 = 3.46.
Applying the skills
Question 10. Light travels at 3 × 108 m/s. A light signal travels 1.5 × 1011 m. How long does it take, in seconds? Give your answer in standard form.
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Time = distance ÷ speed = (1.5 × 1011) ÷ (3 × 108). Divide the numbers: 1.5 ÷ 3 = 0.5. Subtract the powers: 11 − 8 = 3. That gives 0.5 × 103. Since 0.5 is less than 1, rewrite it as 5 × 10−1, so the answer is 5 × 10−1+3 = 5 × 102 seconds.
Check: 500 seconds × 3 × 108 m/s = 1.5 × 1011 m.
Question 11. Estimate (8.2 × 103) × (1.9 × 105) by rounding to one significant figure. Then find the exact value, to 3 significant figures.
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Estimate: 8 × 2 = 16 and 103+5 = 108, so about 16 × 108 = 1.6 × 109.
Exact: 8.2 × 1.9 = 15.58, so the product is 15.58 × 108 = 1.558 × 109. To 3 significant figures: 1.56 × 109. The estimate is close, so the answer is sensible.
Question 12. Find x if 4x = 1/8.
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Write both sides as powers of 2: 4x = (22)x = 22x, and 1/8 = 2−3. So 2x = −3, which gives x = −3/2.
Check: 4−3/2 = 1/(√4)3 = 1/23 = 1/8.
If you got these wrong
| What went wrong | Likely cause | Go back to |
|---|---|---|
| Q1 to Q4: wrong sign or wrong size | Treating a negative power as a negative number, or mixing up the root and the power | Apply index laws with negative powers |
| Q5, Q6: power is one out | Counting zeros instead of places, or forgetting to re-standardise 15 × 103 | Convert a small measurement to standard form |
| Q7, Q10: wrong order or wrong power | Comparing the decimal parts before the powers, or adding powers when dividing | Compare quantities with different powers of ten |
| Q8, Q9: stray number left in front | Taking out the square factor without square-rooting it | Simplify a root using square factors |
| Q11: no sense check | Not rounding first, or trusting a calculator entry | Estimate the size of a standard-form answer |
For a wider view of how these skills fit together, return to the module overview. The non-calculator working trainer offers more practice at exact working by hand.
If you keep making the same kind of slip even after reading the explanation, it may be a habit rather than a gap in knowledge. That is where online one-to-one Mathematics tuition can help, because a teacher sees the working as you write it.